# Aichelburg, Sexl ## On the gravitational field of a massless particle \[Links: [Inspire](https://inspirehep.net/literature/61172), [DOI](https://link.springer.com/article/10.1007/BF00758149)\] \[Abstract: The gravitational field of a massless point particle is first calculated using the linearized field equations. The result is identical with the exact solution, obtained from the Schwarzschild metric by means of a singular Lorentz transformation. The gravitational field of the particle is nonvanishing only on a plane containing the particle and orthogonal to the direction of motion. On this plane the Riemann tensor has a $\delta$-like singularity and is exactly of Petrov type $N$.\] ## Summary - this is the original paper on obtaining the [[0117 Shockwave|shockwave]] solution in flat space - boosting a black hole to the null limit gives the exact shockwave solution # Carter ## Axisymmetric Black Hole Has Only Two Degrees of Freedom \[Links: [DOI](https://doi.org/10.1103/PhysRevLett.26.331)\] \[Abstract: A theorem is described which establishes the claim that in a certain canonical sense the Kerr metrics represent "the" (rather than merely "some possible") exterior fields of black holes with the corresponding mass and angular-momentum values.\] ## Summary - [[0455 Black hole uniqueness theorems|uniqueness theorem]] for rotating black holes # Gutzwiller ## Periodic Orbits and Classical Quantization Conditions \[Links: [DOI](https://doi.org/10.1063/1.1665596)\] \[Abstract: The relation between the solutions of the time‐independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of _h_ by half the stability angle times _ℏ_. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times _ℏ_, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.\] # Kosmann ## Dérivées de Lie des spineurs \[Links: [DOI](https://link.springer.com/article/10.1007/BF02428822)\] \[Abstract: The notion of a Lie derivative is generalized to spinor fields on a Spin manifold and the problem of transforming a spinor field under a one-parameter group of diffeomorphisms of the manifold is studied. The invariance of a differential system acting on spinors (Dirac's equations) is studied using methods similar to those used in the case of tensors (Maxwell's equations).\] ## Summary - gives a definition of [[0527 Lie derivative of spinor fields|Lie derivatives for spinor fields]] ## Definition - $\mathcal{L}_X \psi:=X^a \nabla_a \psi-\frac{1}{8} \nabla_{[a} X_{b]}\left[\gamma^a, \gamma^b\right] \psi$ - this differs from the earlier version of [[Lichnerowicz1963]] by the explicit antisymmetrisation - this works for general vector fields rather than just isometries